Wedderburn–Artin Theorem

In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring ''R'' is isomorphic to a product of finitely many -by- matrix rings over division rings , for some integers , both of which are uniquely determined up to permutation of the index . In particular, any simple left or right Artinian ring is isomorphic to an ''n''-by-''n'' matrix ring over a division ring ''D'', where both ''n'' and ''D'' are uniquely determined.

Theorem

Let be a (Artinian) semisimple ring. Then the Wedderburn–Artin theorem states that is isomorphic to a product of finitely many -by- matrix rings $M_(D_i)$ over division rings , for some integers , both of which are uniquely determined up to permutation of the index .

There is also a version of the Wedderburn–Artin theorem for algebras over a field . If is a finite-dimensional semisimple -algebra, then each in the above statement is a finite-dimensional division algebra over . The center of each need not be ; it could be a finite extension of .

Note that if is a finite-dimensional simple algebra over a division ring ''E'', ''D'' need not be contained in ''E''. For example, matrix rings over the complex numbers are finite-dimensional simple algebras over the real numbers.

Proof

There are various proofs of the Wedderburn–Artin theorem. A common modern one takes the following approach.

Suppose the ring $R$ is semisimple. Then the right $R$-module $R_R$ is isomorphic to a finite direct sum of simple modules (which are the same as minimal right ideals of $R$). Write this direct sum as

: $R_R \;\cong\; \bigoplus_^m I_i^$

where the $I_i$ are mutually nonisomorphic simple right $R$-modules, the th one appearing with multiplicity $n_i$. This gives an isomorphism of endomorphism rings

: $\mathrm(R_R) \;\cong\; \bigoplus_^m \mathrm\big(I_i^\big)$

and we can identify $\mathrm\big(I_i^\big)$ with a ring of matrices

: $\mathrm\big(I_i^\big) \;\cong\; M_\big(\mathrm(I_i)\big)$

where the endomorphism ring $\mathrm(I_i)$ of $I_i$ is a division ring by Schur's lemma, because $I_i$ is simple. Since $R \cong \mathrm(R_R)$ we conclude

: $R \;\cong\; \bigoplus_^m M_\big(\mathrm(I_i)\big) \,.$

Here we used right modules because $R \cong \mathrm(R_R)$; if we used left modules $R$ would be isomorphic to the opposite algebra of $\mathrm(_R R)$, but the proof would still go through. To see this proof in a larger context, see Decomposition of a module. For the proof of an important special case, see Simple Artinian ring.

Consequences

Since a finite-dimensional algebra over a field is Artinian, the Wedderburn–Artin theorem implies that every finite-dimensional simple algebra over a field is isomorphic to an ''n''-by-''n'' matrix ring over some finite-dimensional division algebra ''D'' over $k$, where both ''n'' and ''D'' are uniquely determined. This was shown by Joseph Wedderburn. Emil Artin later generalized this result to the case of simple left or right Artinian rings.

Since the only finite-dimensional division algebra over an algebraically closed field is the field itself, the Wedderburn–Artin theorem has strong consequences in this case. Let be a semisimple ring that is a finite-dimensional algebra over an algebraically closed field $k$. Then is a finite product $\textstyle \prod_^r M_(k)$ where the $n_i$ are positive integers and $M_(k)$ is the algebra of $n_i \times n_i$ matrices over $k$.

Furthermore, the Wedderburn–Artin theorem reduces the problem of classifying finite-dimensional central simple algebras over a field $k$ to the problem of classifying finite-dimensional central division algebras over $k$: that is, division algebras over $k$ whose center is $k$. It implies that any finite-dimensional central simple algebra over $k$ is isomorphic to a matrix algebra $\textstyle M_(D)$
where $D$ is a finite-dimensional central division algebra over $k$.

See also

* Maschke's theorem
* Brauer group
* Jacobson density theorem
* Hypercomplex number
* Emil Artin
* Joseph Wedderburn

References

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{{DEFAULTSORT:Artin-Wedderburn Theorem
Theorems in ring theory